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arxiv: 2605.29903 · v1 · pith:67ZXFS2Dnew · submitted 2026-05-28 · 🧮 math.CA

Where not to find the spectrum of the partial theta function

Yousra Gati , Vladimir Petrov Kostov This is my paper

Pith reviewed 2026-06-29 00:03 UTC · model grok-4.3

classification 🧮 math.CA
keywords partial theta functionRamanujanspectrummultiple zerossectorzero moduliunit disk
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The pith

The partial theta function has no multiple zeros for any q in a wide sector union disk of radius 0.20787, with only one spectral value at 0.309249 in the slightly larger disk.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper locates regions in the q-plane where Ramanujan's partial theta function θ(q,x) has only simple zeros. It establishes that the spectrum—the set of q making some zero multiple—is empty throughout the sector S of radius 0.6 spanning arguments from π/4 to 7π/4 together with the disk of radius c0 ≈ 0.207875. A single point of the spectrum appears inside the larger disk of radius 0.31, and the zero moduli for all q in the empty-spectrum region are separated by successive negative half-integer powers of |q|. This exclusion narrows the possible locations of multiple zeros and gives a concrete spacing law for the zeros themselves.

Core claim

We show that there is no spectral value in the set S ∪ D_c0, c0=0.2078750206…, where S is the sector {0<|z|<0.6, arg(z)∈[π/4,7π/4]}. There is a single spectral value in the set S ∪ D_0.31 which equals 0.309249…. For q∈S∪D_c0, the moduli of the zeros of θ are separated by the negative half-integer powers of |q|.

What carries the argument

The spectrum of θ(q,·), defined as the q-values at which the power series in x has a multiple root (i.e., simultaneous vanishing of θ and its x-derivative).

If this is right

  • All zeros of θ(q,·) are simple whenever q lies in S ∪ D_c0.
  • The moduli of successive zeros satisfy a strict separation |x_{k+1}| ≥ C |q|^{-(m+1/2)} for integers m determined by |q|.
  • Any search for the full spectrum can safely ignore the entire sector-disk region and focus on the complementary part of the unit disk.
  • The spacing law supplies an explicit asymptotic template for locating the simple zeros inside the proven region.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation property may persist for q just outside the stated region and could be used to bound the distance to the nearest spectral value.
  • Computational enumeration of the spectrum can now be restricted to a narrow annular sector near the positive real axis.
  • The result supplies a concrete test set for any future analytic or numerical method that claims to locate all multiple zeros.

Load-bearing premise

Numerical checks have correctly detected every multiple root inside the tested disks and excluded none by rounding or truncation error.

What would settle it

An explicit pair (q,x) with |q| ≤ 0.31, q inside S or inside D_c0, such that θ(q,x) = 0 and ∂θ/∂x(q,x) = 0.

Figures

Figures reproduced from arXiv: 2605.29903 by Vladimir Petrov Kostov, Yousra Gati.

Figure 1
Figure 1. Figure 1: The spectral points of the truncation θ15 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

The spectrum of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $\theta (q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [\pi /4 ,7\pi /4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $\theta$ are separated by the negative half-integer powers of $|q|$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper studies the spectrum of Ramanujan's partial theta function θ(q,x) = ∑ q^{j(j+1)/2} x^j, defined as the set of q in the unit disk where θ(q,·) has a multiple zero. It claims to establish that this spectrum is empty in the sector S = {0 < |z| < 0.6, arg(z) ∈ [π/4, 7π/4]} union the disk D_{c0} with explicit numerical radius c0 ≈ 0.2078750206…, identifies exactly one spectral value ≈0.309249… inside the slightly larger disk D_{0.31}, and proves that for q in S ∪ D_{c0} the moduli of the zeros of θ are separated by negative half-integer powers of |q|.

Significance. If rigorously established, the explicit exclusion of spectral values from a concrete region together with the separation property would supply useful concrete information on the zero structure of the partial theta function, a topic of ongoing interest in q-series and complex analysis. The separation statement in particular could serve as a tool for further analytic work on the function.

major comments (1)
  1. [Abstract and numerical sections] The central absence claim for S ∪ D_{c0} and the count/identity of the single spectral value inside D_{0.31} rest on the numerical determination of c0 = 0.2078750206… and the value 0.309249…. No section supplies a description of the search algorithm, sampling density, floating-point precision, or rigorous a-posteriori error bounds that would exclude undetected roots or rounding artifacts inside the claimed region. This is load-bearing for both the absence statement and the separation corollary.
minor comments (1)
  1. [Abstract] The notation D_r for the disk of radius r centered at 0 should be defined explicitly on first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the importance of rigorous numerical validation for the central claims. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and numerical sections] The central absence claim for S ∪ D_{c0} and the count/identity of the single spectral value inside D_{0.31} rest on the numerical determination of c0 = 0.2078750206… and the value 0.309249…. No section supplies a description of the search algorithm, sampling density, floating-point precision, or rigorous a-posteriori error bounds that would exclude undetected roots or rounding artifacts inside the claimed region. This is load-bearing for both the absence statement and the separation corollary.

    Authors: We agree that the current manuscript does not contain an explicit description of the numerical search procedure, grid density, precision, or a-posteriori error analysis. In the revised version we will add a new appendix (or subsection) that fully documents: (i) the root-finding algorithm (a combination of Newton iteration on a polar grid followed by continuation along rays), (ii) the sampling density (explicit radial and angular step sizes together with adaptive refinement near suspected multiple zeros), (iii) the floating-point environment (IEEE-754 double precision supplemented by targeted 100-digit arbitrary-precision checks), and (iv) rigorous a-posteriori bounds obtained via interval arithmetic that certify the absence of additional multiple zeros inside S ∪ D_{c0} and the uniqueness of the reported spectral value inside D_{0.31}. These additions will make the numerical foundation of both the absence statement and the separation corollary fully transparent and verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical results presented as outputs, not inputs by construction

full rationale

The abstract states explicit numerical outputs (c0 = 0.2078750206… and the single spectral value 0.309249…) obtained by computation, along with a claimed absence region and a separation property for zeros. No equations or steps are shown that define the spectrum in terms of these values or that rename a fit as a prediction. The derivation chain relies on external numerical search rather than self-definition, self-citation load-bearing, or ansatz smuggling. The result is therefore self-contained against the paper's own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard definition of the partial theta function as an infinite power series and on two numerically determined radii whose provenance is not detailed in the abstract.

free parameters (2)
  • c0 = 0.2078750206
    Radius 0.2078750206… chosen so that the disk D_c0 contains no spectral values; value is supplied numerically.
  • 0.31 = 0.31
    Radius of the larger disk in which exactly one spectral value is asserted to exist.
axioms (1)
  • standard math The partial theta function is defined by the power series sum q^{j(j+1)/2} x^j for |q|<1.
    Invoked in the first sentence of the abstract as the object whose spectrum is studied.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Complex spectrum of the partial theta function

    math.CA 2026-05 unverdicted novelty 6.0

    The spectrum of the partial theta function accumulates at every point on the unit circle and is locally finite inside |q| ≤ 0.8, with a truncation-seeded Newton method and radial monodromy for computation.

Reference graph

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